The side angle side formula which is also known as the SAS formula is used to calculate the area of the triangle using trigonometry rules. This formula which is based on the side angle side theorem helps us to calculate the area of a triangle. As the name suggests, side angle side represents the two sides and the angle between them. Let us explore more about the side angle side formula to calculate the area of a triangle.

## What is Side Angle Side Formula?

The side angle side formula is the SAS area formula which means we can find the area of a triangle if the length of two sides of a triangle and its included angle is known. The SAS formula is expressed as:

Area of a triangle = (1/2) × side_{1} × side_{2} × sin (included angle)

The side angle side formula was created on the basis of the Side Angle Side Theorem. According to the **Side Angle Side theorem**, two triangles are said to be congruent if two sides and the angle that lies between these two sides are equal. The angle between the two sides is known as the included angle.

### Derivation of Side Angle Side Formula

We know that the area of a triangle is = 1/2 × base × height. So, let us consider the following triangle to understand the derivation of the SAS formula using the steps given below.

- Step 1: If the two given sides are ‘a’ and ‘b’ and the included angle between them is ‘c’.
- Step 2: If we draw a perpendicular ‘p’ from X to side YZ, then using the trigonometric ratio, we can write the value of p as, p = a × sin c considering p as the height, and applying the formula Sin c = p/a.
- Step 3: Since we know that the area of a triangle = 1/2 × base × height. Substituting the value of base as b and height as p, the area of the triangle will be = 1/2 × b × p.
- Step 4: Since p = a × sin c, the formula for the area of the triangle will be = 1/2 × b × a × sin c

Therefore, the side angle side formula or the area of the triangle using the SAS formula = 1/2 × a × b × sin c

Let us work on some problems to understand the side angle side formula.

## Examples Using Side Angle Side Formula

**Example 1:** What will be the area of a triangle whose sides are of length 5 cm and 10 cm and its included angle is 30°?

**Solution:**

We know that the side angle side formula is given as:

Area of a triangle = (1/2) × side_{1} × side_{2} × sin (included angle)

Given: side_{1} = 5 cm, side_{2} = 10 cm, sin (included angle) = sin 30° = 1/2

Substituting the values,

Area = (1/2) × 5 × 10 × sin 30°

= (1/2) × 5 × 10 × (1/2)

= 12.5 cm^{2}

**Answer: **Thus, the area of the triangle is 12.5 cm^{2}

## Examples Using Side Angle Side Formula

**Example 1:** What will be the area of a triangle whose sides are of length 5 cm and 10 cm and its included angle is 30°?

**Solution:**

We know that the side angle side formula is given as:

Area of a triangle = (1/2) × side_{1} × side_{2} × sin (included angle)

Given: side_{1} = 5 cm, side_{2} = 10 cm, sin (included angle) = sin 30° = 1/2

Substituting the values,

Area = (1/2) × 5 × 10 × sin 30°

= (1/2) × 5 × 10 × (1/2)

= 12.5 cm^{2}

**Answer: **Thus, the area of the triangle is 12.5 cm^{2}

**Example 2:** In the triangle shown below, find all the dimensions using the side angle side formula.

**Solution:**

Given: Angle A = 49°, b = 5, c = 7

To find: a, Angle B, Angle C

Let us name side AB as c, and side AC as b

Let us follow the steps of the side angle side formula:

Step 1: To find the value of ‘a’ let us use the Law of Cosines

a^{2} = b^{2} + c^{2} − 2bc cosA

a^{2} = 5^{2} + 7^{2} − 2 × 5 × 7 × cos(49°)

a^{2} = 25 + 49 − 70 × cos(49°)

a^{2} = 74 − 70 × 0.6560…

a^{2} = 74 − 45.924… = 28.075…

a = √28.075…

a = 5.298..

**a = 5.30** (rounded upto 2 decimal places)

Step 2: To find the value of the smaller angle, we will use the law of sines

Angle B is smaller than Angle C as Angle B is opposite to the shorter side.

Thus, we will choose angle B and apply law of sines,

sin B / b = sin A / a

sin B / 5 = sin(49°) / 5.298…

sin B = (sin(49°) × 5) / 5.298…

sin B = 0.7122…

B = sin^{−1}(0.7122…)

**B = 45.4°** (rounded upto 1 decimal place)

Step 3: Now to find angle C, the angle sum property of the triangle can be used

C = 180° − 49° − 45.4°

**C = 85.6°**

**Answer:** Thus, we have calculated all the missing dimensions of the triangle.

## FAQS on Side Angle Side Formula

### What is the Side Angle Side Formula?

The **side angle side formula** which is also known as the SAS formula is used to calculate the area of the triangle using trigonometry rules. The formula is written as, Area of a triangle = (1/2) × side_{1} × side_{2} × sin (included angle), which means that if the two sides and the angle included between them is given then the area of the triangle can be calculated using the given formula.

### What is the SAS Theorem in Geometry?

According to the **Side Angle Side theorem**, two triangles are said to be congruent if two sides and the angle that lies between these two sides are equal. The angle between the two sides is known as the included angle.

### What is the Use of the Side Angle Side Formula?

The side angle side formula can be used to find the area of a triangle when the two sides and the included angle is given. The other uses of the formula are that we can find the hypotenuse, the unknown side of a right-angled triangle using the trigonometric law of Cosine. We can use the law of sines to find the smaller angle, and then the third angle of the triangle can be calculated since two angles are known.

## Solving SAS Triangles

*“SAS” means “Side, Angle, Side”*

“SAS” is when we know two sides and the angle between them. |

To solve an SAS triangle

- use The Law of Cosines to calculate the unknown side,
- then use The Law of Sines to find the smaller of the other two angles,
- and then use the three angles add to 180° to find the last angle.

Example 1

Now we use the The Law of Sines to find the smaller of the other two angles.

**Why the smaller angle?** Because the inverse sine function gives answers less than 90° even for angles greater than 90°. By choosing the smaller angle (a triangle won’t have two angles greater than 90°) we avoid that problem. Note: the smaller angle is the one facing the shorter side.

Choose angle B:

sin B / b = sin A / a

sin B / 5 = sin(49°) / 5.298…

Did you notice that we didn’t use **a = 5.30**. That number is rounded to 2 decimal places. It’s much better to use the unrounded number 5.298… which should still be on our calculator from the last calculation.

sin B = (sin(49°) × 5) / 5.298…

sin B = 0.7122…

B = sin^{−1}(0.7122…)

B = **45.4°** to one decimal place

Now we find angle C, which is easy using ‘angles of a triangle add to 180°’:

C = 180° − 49° − 45.4°

C = **85.6°** to one decimal place

Now we have completely solved the triangle i.e. we have found all its angles and sides.

Now for The Law of Sines.

Choose the smaller angle? We don’t have to! Angle R is greater than 90°, so angles P and Q must be less than 90°.

sin P / p = sin R / r

Mastering this skill needs lots of practice, so …

### Congruence of Triangles

The congruence of two or more triangles depends on the measure of the sides and angles. The three sides of a triangle determine the size and the three angles of the triangle determine the shape. Two triangles are said to be congruent if their corresponding pairs of sides and corresponding angles are equal. They are the same shape and size. These triangles can be slid, rotated, flipped, and flipped to look the same. If you rearrange them, they match each other. The sign of congruence is “≅”.

In mathematics, congruence means that two figures are similar in shape and size. There are basically four laws of congruence that prove the congruence of two triangles. However, you need to find all six dimensions. Therefore, the congruence of a triangle can be estimated by knowing only three out of six values.

### Congruence in Triangles

Two triangles are said to be congruent if three angles and three sides of a triangle are equal to the corresponding angles and sides of another triangle. From Δ PQR and ΔXYZ, we observe that PQ = XY, PR = XZ, QR = YZ, and ∠P = ∠X, ∠Q = ∠Y, and ∠R = ∠Z. Then we can say ΔPQR ≅ ΔXYZ. Two triangles must be the same size and shape to be congruent. Two triangles to be considered must overlap each other, when you rotate, reflect, or move a triangle, its position or shape will not change.

**Conditions of Congruence in Triangles**

Two triangles are said to be congruent if three angles and three sides of a triangle are equal to the corresponding angles and sides of another triangle. It is not necessary to find all six corresponding elements of two triangles to determine congruence. Studies and experiments have shown that there are five conditions under which two triangles are congruent.

**SSS (Side – Side – Side) Rule**

Two triangles are said to be congruent by the SSS law if three sides of one triangle are equal to the corresponding three sides of the second triangle. In the given figure, AB = PQ, BC = QR, AC = PR,

Therefore, ΔABC ≅ ΔPQR

**SAS (Side – Angle – Side) Rule**

Two triangles are said to be SAS congruent if the angle between two sides of a triangle is equal to the angle between two sides of the second triangle. In the given figure, the sides are AB = PQ, AC = PR, and the angle between AC and AB is equal to the angle between PR and PQ. That is, ∠A = ∠P. So, ∆ABC ≅ ∆PQR.

**ASA (Angle – Side – Angle) Rule**

According to the ASA criteria, two triangles are congruent if two angles of one triangle and the side between them are equal to the corresponding angles of another triangle and the side between them.

In the above figure, ∠ B = ∠ Q, ∠ C = ∠ R, and the sides between ∠ B and ∠ C, ∠ Q and ∠ R are equal. That is, BC = QR. So ∆ABC ≅ ∆PQR.

**AAS (Angle – Angle – Side) Rule**

According to the AAS criterion, two triangles are congruent if two angles of a triangle and a side not between them are equal to the corresponding angle of the other triangle and the sides that are not between them are equal. According to the AAS criterion, if ΔABC ≅ ΔXYZ, then the third angle (∠ABC) and the other two sides (AC and BC) ΔABC must be equal to that angle (∠XYZ) and sides (XZ and YZ) ΔXYZ.

**RHS (Right angle- Hypotenuse-Side) Rule**

Two right triangles are RHS congruent if the hypotenuse and side of a right triangle are equal to the hypotenuse and side of the second right triangle. In the figure above, the hypotenuse is XZ = RT and the side is YZ = ST, so ∆XYZ ≅ ∆RST.

### Side Angle Side

SAS congruence is a term also known as lateral angle congruence used to describe the relationship between two congruent shapes. Let’s take a closer look at SAS triangle congruence to understand what SAS means.

These two triangles are the same size and shape. So we can say that they match. They can be seen as examples of congruent triangles. We can represent this in mathematical form using the congruent triangle (≅) symbol, (ΔDEF ≅ ΔPQR). That is, D corresponds to P, E corresponds to Q, and F corresponds to R. ED corresponds to PQ, EF to QR, and DF to PR. Therefore, we can conclude that the corresponding parts of the congruent triangle are equal.

### SAS (Side Angle Side) Formula

The side angle formula, also known as the SAS formula, is used to calculate the area of a triangle using the rules of trigonometry. This formula, based on the side angle theorem, helps you calculate the area of a triangle. As the name suggests, the side angle side refers to two sides and the angle between them. Let’s learn more about the angle formula to find the area of a triangle.

**Derivation of SAS (Side Angle Side) Formula**

We know that the area of a triangle is given as area = 1/2 × base × height. So let us consider the given triangle to understand the derivation of the SAS formula using the steps given below.

**Step 1:** Two sides “a” and “b”, the angle between them “c” are given.

**Step 2: **Draw ‘p’ perpendicular from X to side YZ and use a trigonometric relationship to write the value of p as p = a × sin c. Assume that p is the height and apply the formula. Sin c = p/a.

**Step 3:** Because we know that the area of the triangle = 1/2 × base × height. Substituting the value of the base into b and the height of p, the area of the triangle is = 1/2 × b × p.

**Step 4:** Since p = a × sin c, applying the formula for the area of the triangle is = 1/2 × b × a × sin c.

So, SAS Formula, area = 1/2 × a × b × sin c.

### Sample Questions

**Question 1: What is SAS Principle?**

**Solution:**

The SAS theorem says that two triangles are congruent when the two sides and the angles between them are equal. This requirement does not make the triangles similar but congruent.

**Question 2: What is the side angle formula used for? **

**Answer: **

If you know two sides and the angle between them, you can use the angle of change formula to find the area of a triangle. Another use of the formula is that you can use the trigonometric law of cosines to find the hypotenuse, which is the unknown side of a right triangle. We can use the law of sines to find the smaller angle, and then, knowing the two angles, we can calculate the third angle of the triangle.

**Question 3: What is the area of a triangle when three sides and an angle are given?**

**Answer: **

Given the sides of a triangle along with the angles between them, the area of the triangle can be calculated by the formula

Area = (ab × sin C)/2, where “a” and “b” are the two given sides and C is the angle between them. This method is also known as the side angle method. For example, if two sides of a triangle are 9 and 11 units, and the angle between the two sides is 60°, then area = (11× 9 × sin 60°)/2 = 42.86 units

^{2}.

**Question 4: How to find the perimeter of the SAS triangle?**

**Answer: **

The perimeter of a triangle is defined as the total length of the triangle boundaries. That is, perimeter = sum of all lengths of the sides of a triangle.

Given a triangle ABC with two known sides AB and AC, and an angle ∠BAC between the two sides. Let b be the length of AB, b be the length of AC, and c be the length of BC, then the perimeter can be given as perimeter = a + b + c. Since c is unknown, it can be obtained from the cosine formula (the law of cosines or the law of cosines) c

^{2}= a^{2}+ b^{2}2ab cos(∠BAC). So the perimeter of the triangle SAS = a + b + √[a^{2 }+ b^{2 }− 2abcos(∠BAC)] .

**Question 5: What is the area of a triangle with sides of 15 cm and 18 cm and the angle between them is 45°?**

**Solution:**

By Side Angle Side Formula,

Area of triangle = 1/2 × a × b × sin c

Given, a = 15cm

b = 18 cm

Area = 1/2 × 15 × 18 × sin 45°= 95.47 cm

^{2}.

**Question 6: What is the area of a triangle with sides of 5 cm and 10 cm and the angle between them is 30°?**

**Solution:**

The formula for one side is,

Triangle area = (1/2) × side

_{1}× side_{2}× sin (including angle)Given, side

_{1 }= 5cm, side_{2}= 10cm, sin(including angle) = sin 30° = 1/2. Substitute values,Area = (1 /2) × 5 × 10 × sin 30°

= (1/2) × 5 × 10 × (1/2)

= 12.5 cm

^{2}.So, the area of the triangle is 12.5 cm

^{2}.

**Question 7: If EA = 30 units, ED = 20 units, and ∠AED = 30°, find the area of the quadrilateral ADCB. Also, B divides EA by 1:2 and C is the midpoint of ED. **

**Solution: **

It is given that sides,

EA = 40 units and ED = 30 units

Also, angle between EA and ED = ∠AED = 30°

By side angle side formula, Area of ΔAED = 1/2 × EA × ED × sin (30°) = 1/2 × 40 × 30 × sin (30°) = 300 units

^{2}As B divides EA by 1:2

3EB = AE

EB = AE/3 = 40 / 3 = 13.3 units

Given that C is the midpoint of ED , thus EC = ED/2 = 30/2 = 15 units

Now area of ΔECB = 1/2 × EB × EC × sin (30°) =1/2 × 13.3 × 15 × sin (30°) = 49.875 units

^{2}Area (ADCB) = Area (ΔADE) – Area (ΔBCE) = 300 – 49.875 = 250.125 unit

^{2}Area (ADCB) = 250.125 unit

^{2}.

### Solved example

**Question:** What will be the area of a triangle whose sides are of length 6 cm and 10 cm and its included angle is 45 degrees ?

**Solution:**

Using the formula:

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